If p is a permutation of the atoms of i.n,
then p is said to be a permutation vector of
order n,and if n=#b,
then p{b is a permutation of the items of b .

C.p yields a list of boxed lists of the atoms
of i.#p,called the standard cycle
representation of the permutation p .Thus,
if p=:4 5 2 1 0 3,
then C.p is (,2);4 0;5 3 1
because the permutation p moves to
position 2 the item 2,
to 4 the item 0,
to 0 the item 4,
to 5 the item 3,
to 3 the item 1,and
to 1 the item 5.The monad C.
is self-inverse; applied to a standard cycle it gives the
corresponding direct representation.

A given permutation could be represented by cycles in
a variety of ways; the standard form is made unique by
the following restrictions: the cycles are disjoint and exhaustive
(i.e., the atoms of the boxed elements together form a permutation vector);
each boxed cycle is rotated to begin with its largest element;
and the boxed cycles are put in ascending order on their
leading elements.

C. is extended to non-negative non-standard cases
by treating any argument q as a representation of a
permutation of order 1+>./; q .

The monad C.!.2 computes the parity of a
permutation p ;it is 1 or _1
as the number is even or odd of pairwise interchanges necessary
to get p from the identity permutation i.#p
(and 0 if p is not a permutation).
For example:

] x=: 2 , (i.4) ,: 1 0 2 3
2 2 2 2
0 1 2 3
1 0 2 3
C.!.2 x
0 1 _1

If p and c are standard and cycle representations
of order #b,then p C. b and c C. b produce
the permutation of b .The arguments p and c
can be non-standard in ways to be defined. In particular,
negative integers down to -#b may be used, and are treated
as their residues modulo #b .

If q is not boxed, and the elements of (#b)|q
are distinct, then q C. b is equivalent to p{b ,
where p is the standard form of q
that is given by p=:((i.n)-.n|q),n|q ,
for n=:#b . In other words, positions occurring in q
are moved to the tail end.

If q is boxed, the elements of (#b)|>j{q must
be distinct for each j ,and the boxes are applied
in succession. For example: